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What is the probability of one robot going to the 1st

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Unformatted text preview: probably allocated to each of the n floors, the distinct patterns are not equally likely (Why? For example, if there are 3 floors and 2 robots, what is the probability of all robots going to the 1st floor? What is the probability of one robot going to the 1st floor and the other robot going to the 2nd floor?). The correct probability should be 1 . nr Problem 6 Problem 6. In an experiment k balls are drawn randomly without replacement from an urn containing 1 red ball, 2 blue balls and 17 yellow balls. All the balls are identical apart from their colours. 1 Suppose 0 ≤ k ≤ 17. Show how you may obtain the number of distinct combinations of the k balls by considering the identity (1 + x )(1 + x + x 2 )(1 − x )−1 ≡ (1 − x 2 )(1 − x 3 )(1 − x )−3 . 2 Show that there are six distinct ways of drawing three balls randomly without replacement from the urn. Solution to Problem 6 Solution. 1 The number of ways of drawing k balls from the urn is equal to the coefficient of x k in the expression (why? Note that (1 + x )(1 + x + x 2 )(1 − x )−1 = (1 + x )(1 + x + x 2 )(1 + x + x 2 + x 3 + ...). 2 The number of ways of drawing k balls from the urn is equal to the coefficient of x k in the above expression.) Using the expression on the RHS, we get (1 − x 2 )(1 − x 3 )(1 − x )−3 = (1 − x 2 − x 3 + x 5 )(1 + 3x + 6x 2 + 10x 3 + ...). From above, we see that the coefficient of x 3 is −1 − 3 + 10 = 6. Problem 7 Problem 7. (Class test Fall 2008) A lady has three rings and eight distinguishable fingers (excluding the two thumbs). 1 Suppose the three rings are identical. 1 2 3 2 How many distinct ways are there to wear the three rings on the same finger? How many distinct ways are there to wear two rings on the same finger and the third ring on a different finger? How many distinct ways are there to wear the three rings on three separate fingers? Now suppose the three rings are distinguishable. Using part (a) or otherwise, find the total number of distinct ways to wear the three rings on the lady’s eight fing...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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